The search for vacua has been carried out by solving the sets of quadratic equations for the singlets in the tables above.
Once we found solutions, we checked for each candidate vacuum the rank of the embedding tensor, the signature of the resulting Cartan–Killing matrix and the full mass spectrum.
Overall we found 5 different Anti-de Sitter vacua, 5 Minkowski vacua and 2 de Sitter vacua.
The vacua with negative cosmological constant are all pertaining to the same gauging, namely the maximal SO(6) theory of [1]}, and were all already known [2]}, [3]}, [4]}.
Among the Minkowski vacua there are the Cremmer–Scherk–Schwarz gaugings [5]}, [6]} with various mass parameters and a supersymmetric vacuum for the SO\(^*\) (6) theory discovered in [7]}, but we also find three new vacua with a non-abelian gauge group (like those in [8]} for the analogous analysis of maximal supergravity in 4 dimensions).
Finally, we also find 2 de Sitter vacua, resulting from gauging of the semisimple groups SO(3,3) [9]} and SO\(^*(6)\) , the latter being new.
All the vacua are reported in the table 7, together with the number of supersymmetry they preserve, the original gauging, the residual gauge group and the reference where they were first discovered.
In the appendix we provide for each vacuum one instance of fermion shift values reproducing the critical point mentioned in the table.
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